Abstract

We consider a random walk with death in [−N, N] moving in a time dependent environment. The environment is a system of particles which describes a current ﬂux from N to −N. Its evolution is inﬂuenced by the presence of the random walk and in turns it aﬀects the jump rates of the random walk in a neighborhood of the endpoints, determining also the rate for the random walk to die. We prove an upper bound (uniform in N) for the survival probability up to time t which goes as c exp{−bN−2 t}, with c and b positive constants.